HOMOLOGY AND COHOMOLOGY OPERATIONS

John Rognes

1. Based spaces

1.1 Homology and cohomology of spaces. 1.1.1. We will work in the category of cofibrantly based compactly generated weak Hausdorff spaces. 1.1.2. Let all chains, cochains, homology and cohomology groups have implicit coefficients ℤ/2. Hence we briefly write H_{∗}(X) = H_{∗}(X; ℤ/2) and H^{∗}(X) = H^{∗}(X; ℤ/2). 1.1.3. The evaluation of cochains on chains S^{∗}(X)⊗S_{∗}(X) → ℤ/2 induces a perfect pairing ⟨ , ⟩: H^{∗}(X) ⊗ H_{∗}(X) → ℤ/2, or equivalently a natural isomorphism

H^{∗}(X) ≅ Hom(H_{∗}(X), ℤ/2).

This is a case of the universal coefficient theorem. 1.1.4. The Eilenberg–Zilber shuffle homomorphism S_{∗}(X) ⊗ S_{∗}(Y) → S_{∗}(X × Y) provides an associative and commutative chain homotopy inverse to the Alexander– Whitney map S_{∗}(X × Y) → S_{∗}(X) ⊗ S_{∗}(Y). Both maps are chain homotopy equivalences, and induce the Künneth isomorphisms

H_{∗}(X) ⊗ H_{∗}(Y) ≅ H_{∗}(X × Y)

and

H^{∗}(X) ⊗ H^{∗}(Y) ≅ H^{∗}(X × Y).

We write x × y for the image of x ⊗ y under either of these isomorphisms. See [McL]. 1.1.5. The diagonal map Δ: X → X × X and the collapse map c: X → ∗ induce a coproduct

ψ: H_{∗}(X) ⟶^{Δ_{∗}} H_{∗}(X × X) ≅ H_{∗}(X) ⊗ H_{∗}(X)

and counit

ϵ: H_{∗}(X) ⟶^{c_{∗}} H_{∗}(∗) = ℤ/2

which give H_{∗}(X) the structure of a coalgebra. It is cocommutative, and coaugmented by η: ℤ/2 = H_{∗}(∗) → H_{∗}(X) induced by the inclusion i: ∗ → X of the base point. 1.1.6. Dually H^{∗}(X) ≅ Hom(H_{∗}(X), ℤ/2) is an algebra, with cup product

ϕ: H^{∗}(X) ⊗ H^{∗}(X) → H^{∗}(X)

Typeset by AMS-TEX

derived from Δ^{∗} and unit η: ℤ/2 → H^{∗}(X) derived from c^{∗}. It is commutative, and augmented by ϵ: H^{∗}(X) → ℤ/2 derived from i^{∗}. We write ϕ(x ⊗ y) = x ⌣ y = xy for the cup product. 1.1.7. Our algebras, coalgebras and Hopf algebras are always associative and unital (or coassociative and counital, etc.).

1.2. The Steenrod squares. 1.2.1. Let C_{2} = {1, T } be the group of order 2, and let EC_{2} be a free contractible C_{2} -space. Its orbit space BC_{2} = EC_{2}/C_{2} is the classifying space for principal C_{2} -bundles. One model for EC_{2} is S^{∞} = ⋃_{n} S^{n} with the antipodal action. Let ϵ: W_{∗} → ℤ/2 be the periodic resolution of ℤ/2 by free ℤ/2[C_{2}]-modules. Specifically, let W_{i} = ℤ/2[C_{2}]{e_{i}} be the free module of rank 1 on a generator e_{i} and let ∂(e_{i+1}) = 1 · e_{i} + T · e_{i}, for all i ≥ 0. Then H_{i}(BC_{2}) ≅ H_{i}(W_{∗} ⊗_{C_{2}} ℤ/2) = ℤ/2{e_{i}} for all i ≥ 0. Dually H^{∗}(BC_{2}) = ℤ/2[t], with t ∈ H^{1}(BC_{2}) dual to e_{1} . Then t^{n} is dual to e_{n}. 1.2.2. There is a chain homotopy equivalence H_{∗}(X) → S_{∗}(X), where H_{∗}(X) is viewed as a chain complex with zero boundary maps, given by choosing a basis for H_{∗}(X) and choosing representing cycles in Z_{∗}(X) ⊂ S_{∗}(X) for these homology classes. Likewise there is a C_{2} -equivariant chain homotopy equivalence W_{∗} → S_{∗}(EC_{2}). Viewing EC_{2} as S^{∞} as the union of the S^{n} ’s with S^{n} = S^{n−1} ∪ D_{+}^{n} ∪ D_{−}^{n}, one model takes e_{n} and T e_{n} to chains representing D_{+}^{n} and D_{−}^{n}, respectively. 1.2.3. We can now construct the Steenrod squaring operations. The diagonal Δ induces a C_{2} -equivariant map 1 × Δ: EC_{2} × X → EC_{2} × X^{2} . Here C_{2} acts by T (e, x) = (T e, x) on the left, and by T (e, x, y) = (T e, y, x) on the right. Hence there is an induced map of orbit spaces

1 ×_{C_{2}} Δ: BC_{2} × X → EC_{2} ×_{C_{2}} X^{2}.

We can compute the cohomology of the target. There are isomorphisms

H^{∗}(EC_{2} ×_{C_{2}} X^{2}) ≅ H^{∗}(S_{∗}(EC_{2}) ⊗_{C_{2}} S_{∗}(X)^{⊗2}) ≅ H^{∗}(W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2})

induced by shuffle homomorphisms, by the chain equivalence W_{∗} ≃ S_{∗}(EC_{2}), and the chain equivalence S_{∗}(X) ≃ H_{∗}(X). Hence we get a homomorphism

θ^{∗}: H^{∗}(W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2}) ⟶ H^{∗}(BC_{2} × X)

by composing the isomorphisms above with (1 ×_{C_{2}} Δ)^{∗}. 1.2.4. Now suppose we are given a class x ∈ H^{n}(X). View it as an n-cocycle ξ: H_{n}(X) → ℤ/2. Also view the augmentation ϵ: W_{∗} → ℤ/2 as a 0-cocycle. The product ϵ ⊗ ξ ⊗ ξ: W_{∗} ⊗ H_{∗}(X)^{⊗2} → ℤ/2 is C_{2} -equivariant, hence descends to define a cohomology class

[ϵ ⊗ ξ ⊗ ξ] ∈ H^{∗}(W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2}).

By the isomorphism above this gives a cohomology class in H^{∗}(EC_{2} ×_{C_{2}} X^{2}). Applying (1 ×_{C_{2}} Δ)^{∗} we obtain a cohomology class

θ^{∗}[ϵ ⊗ ξ ⊗ ξ] ∈ H^{∗}(BC_{2} × X) ≅ H^{∗}(BC_{2}) ⊗ H^{∗}(X).

This class has degree 2n, and can be uniquely written as a sum

∑_{i=0}^{n} t^{n−i} ⊗ Sq^{i}(x) ∈ ⨁_{i=0}^{n} H^{n−i}(BC_{2}) ⊗ H^{n+i}(X)

where the terms Sq^{i}(x) ∈ H^{n+i}(X) are determined by this formula. This defines the Steenrod squaring operation

Sq^{i}: H^{n}(X) → H^{n+i}(X).

1.2.5. We write (i, j) = \binom{i+j}{i} for the binomial coefficient. Thus (i, j) = (j, i). It is (i + j)!/i!j! if i ≥ 0 and j ≥ 0, and 0 if i < 0 or j < 0. In the formulas below, the binomial coefficient needs only be considered mod 2. 1.2.6. Theorem. There are homomorphisms Sq^{i}: H^{∗}(X) → H^{∗}(X) for all n, i ≥ 0, which satisfy:

(1) The Sq^{i} are natural with respect to maps f: X → Y of spaces.

(2) Sq^{i} raises degrees by i.

(3) Sq^{i}(x) = 0 if i > deg(x).

(4) Sq^{i}(x) = x^{2} if i = deg(x).

(5) Sq^{0}(x) = x for all x.

(6) Sq^{1}(x) = β(x) is the Bockstein homomorphism in cohomology associated to the short exact sequence of coefficients 0 → ℤ/2 → ℤ/4 → ℤ/2 → 0.

(7) The external and internal Cartan formulas hold:

Sq^{k}(x × y) = ∑_{i+j=k} Sq^{i}(x) × Sq^{j}(y)

in H^{∗}(X × Y) for x ∈ H^{∗}(X), y ∈ H^{∗}(Y), and

Sq^{k}(xy) = ∑_{i+j=k} Sq^{i}(x)Sq^{j}(y)

in H^{∗}(X) for x, y ∈ H^{∗}(X).

(8) The Adem relations hold: If a < 2b then

Sq^{a}Sq^{b} = ∑_{i}(a − 2i, b − a + i − 1)Sq^{a+b−i}Sq^{i}.

The binomial coefficient can also be written \binom{b−i−1}{a−2i}. The sum runs over i with a − b + 1 ≤ i ≤ a/2.

(9) The Sq^{i} are stable and the Kudo transgression theorem holds: Sq^{i}σ^{∗}(x) = σ^{∗}Sq^{i}(x) where σ^{∗}: H̃^{n+1}(X) → H̃^{n}(ΩX) is the cohomology suspension. Hence also Sq^{i}Σ(x) = ΣSq^{i}(x) where Σ: H̃^{n}(X) → H̃^{n+1}(ΣX) is the suspension isomorphism. If X is simply-connected and if x ∈ H^{n}(ΩX) transgresses to y ∈ H^{n+1}(X) in the Serre spectral sequence of the path space fibration ΩX → P X → X, then Sq^{i}(x) ∈ H^{n+i}(ΩX) transgresses to Sq^{i}(y) ∈ H^{n+i+1}(X). For proofs, see [ES] and [May2].

1.3. The Steenrod algebra A. 1.3.1. Consider sequences I = (i_{1}, …, i_{k}) of integers with i_{s} > 0. Define the degree, length and excess of I by

d(I) = ∑_{s=1}^{k} i_{s},  ℓ(I) = k  and

e(I) = ∑_{s=1}^{k−1}(i_{s} − 2i_{s+1}) + i_{k} = i_{1} − ∑_{s=2}^{k} i_{s}.

The sequence I determines the cohomology operation

Sq^{I} = Sq^{i_{1}} ∘ ⋯ ∘ Sq^{i_{k}}: H^{∗}(X) → H^{∗}(X)

which increases degrees by d(I). I is said to be admissible if i_{s} ≥ 2i_{s+1} for 1 ≤ s < k. The empty sequence I = () is admissible and satisfies d(I) = 0, ℓ(I) = 0 and e(I) = ∞; it determines the identity cohomology operation Sq^{()} = 1. 1.3.2. Define the mod 2 Steenrod algebra A as the quotient of the free (associative, unital) algebra F generated by the Steenrod squaring operations Sq^{i} for i > 0, by the two-sided ideal J generated by the Adem relations 1.2.6(8). Here Sq^{0} = 1. Then J coincides with the two-sided ideal K of elements in F that annihilate H^{∗}(X) for every space X. Hence A equals the algebra generated by the Steenrod squares of cohomology operations on spaces. In turn, every stable mod 2 cohomology operation H^{n}(X) → H^{n+i}(X) is of this form. Hence we may refer to A as precisely the algebra of stable mod 2 cohomology operations. 1.3.3. The admissible monomials Sq^{I} form a ℤ/2-module basis for A. These begin:

1; Sq^{1}; Sq^{2}; Sq^{3}, Sq^{2}Sq^{1}; Sq^{4}, Sq^{3}Sq^{1}; Sq^{5}, Sq^{4}Sq^{1};

Sq^{6}, Sq^{5}Sq^{1}, Sq^{4}Sq^{2}; Sq^{7}, Sq^{6}Sq^{1}, Sq^{5}Sq^{2}, Sq^{4}Sq^{2}Sq^{1}; …

1.3.4. The cohomology of any space X is naturally a left module over the Steenrod algebra A: The pairing

λ: A ⊗ H^{∗}(X) → H^{∗}(X)

takes Sq^{i} ⊗ x to the value Sq^{i}(x), and more generally Sq^{I} ⊗ x to Sq^{I}(x). The cohomology of a space is an unstable left A-algebra, in the sense that Sq^{i}(x) = x^{2} for i = deg(x) and Sq^{i}(x) = 0 for i > deg(x). 1.3.5. The Steenrod algebra A admits a unique structure of Hopf algebra, with coproduct ψ: A → A ⊗ A determined by ψ(Sq^{k}) = ∑_{i+j} Sq^{i} ⊗ Sq^{j}, and counit determined by ϵ(Sq^{i}) = 0 for i > 0. The coproduct is cocommutative. 1.3.6. The cohomology algebra of any space X is naturally a (commutative augmented) left A-module algebra. This means that the cup product map

ϕ: H^{∗}(X) ⊗ H^{∗}(X) → H^{∗}(X)

taking x ⊗ y to xy is a left A-module homomorphism. (And similarly for the unit map η.) Here the left A-module structure on H^{∗}(X) ⊗ H^{∗}(X) uses the Hopf algebra coproduct on A as follows:

a · (x ⊗ y) = ∑ a′x ⊗ a′′y

where ψ(a) = ∑ a′ ⊗ a′′. With a = Sq^{k} and ψ(Sq^{k}) = ∑_{i+j=k} Sq^{i} ⊗ Sq^{j} the assertion that the cup product map is a left A-module homomorphism amounts to the internal Cartan formula 1.2.6(7). 1.3.7. Let X = K(ℤ/2, n) be the Eilenberg–Mac Lane space with π_{n}(X) = ℤ/2 as its only nonzero homotopy group. For any set S let P {S} and E{S} denote the polynomial and exterior algebras over ℤ/2 generated by S, respectively.

1.3.8. Theorem (Serre). Let ι_{n} ∈ H^{n}(K(ℤ/2, n)) = ℤ/2 be the fundamental class, dual to the spherical class in H_{n}(K(ℤ/2, n)) ≅ π_{n}(K(ℤ/2, n)) = ℤ/2. Then

H^{∗}(K(ℤ/2, n)) ≅ P {Sq^{I}(ι_{n}) | I admissible, e(I) < n}.

as an algebra over A.

1.3.9. Let A(n) be the quotient algebra of A by the two-sided ideal generated by Sq^{I} with e(I) > n. Then the admissible monomials Sq^{I} with e(I) ≤ n form a ℤ/2-module basis for A(n). For X = ∏_{n≥0} K(π_{n}(X), n) a product of Eilenberg–Mac Lane spaces such that each π_{n}(X) is a ℤ/2-module, let

D(X) = ⨁_{n≥0} A(n) ⊗ π_{n}(X).

Then H^{∗}(X) is isomorphic to the polynomial algebra on D(X) over ℤ/2, modulo the relations Sq^{i}(x) = x^{2} for i = deg(x).

1.4. The dual Steenrod operations. 1.4.1. By duality, the Steenrod squaring operation Sq^{i}: H^{n}(X) → H^{n+i}(X) induces a homomorphism

Sq_{∗}^{i}: H_{n+i}(X) → H_{n}(X).

(This is clear when X has finite type so H_{∗}(X) ≅ Hom(H^{∗}(X), ℤ/2), and follows in general by naturality.) So

⟨Sq^{i}(x), a⟩ = ⟨x, Sq_{∗}^{i}(a)⟩

for all x ∈ H^{n}(X) and a ∈ H_{n+i}(X). Here ⟨x, c⟩ denotes the Kronecker pairing, evaluating a cohomology class x on a homology class c. 1.4.2. Note that (Sq^{i}Sq^{j})_{∗} = Sq_{∗}^{j}Sq_{∗}^{i}. The dual action makes H_{∗}(X) a right module over the Steenrod algebra

ρ: H_{∗}(X) ⊗ A → H_{∗}(X)

by mapping x ⊗ Sq^{i} to Sq_{∗}^{i}(x), or equivalently a left module over the opposite algebra A^{op} . We remark that since the Steenrod operations are stable, this dual action of Sq_{∗}^{i} commutes with suspensions, and hence also acts on the spectrum homology

H_{n}^{spec}(X) = colim_{k} H_{n+k}(X_{k})

of a spectrum X = (X_{k})_{k}. 1.4.3. We can give H_{∗}(X)⊗H_{∗}(X) a right module action by A using the coproduct ψ. Then H_{∗}(X) is a (cocommutative coaugmented) right A-module coalgebra.

1.5. The dual Steenrod algebra A^{∗}. 1.5.1. The Steenrod algebra A with coproduct ψ is a cocommutative Hopf algebra. Hence the dual

A^{∗} = Hom(A, ℤ/2)

is a commutative Hopf algebra, with coproduct dual to the composition product on A. Its structure was analyzed in [Mil].

1.5.2. Theorem (Milnor). There is an algebra isomorphism

A^{∗} ≅ P {ξ_{1}, ξ_{2}, … }

with deg(ξ_{i}) = 2^{i} − 1 for all i ≥ 1. The coproduct on A^{∗} is given by

ψ(ξ_{k}) = ∑_{i+j=k} ξ_{i}^{2^{j}} ⊗ ξ_{j}.

Here ξ_{0} = 1.

To identify the ξ_{i}, note that A ≅ H_{spec}^{∗}(Hℤ/2; ℤ/2) so A^{∗} ≅ H_{∗}^{spec}(Hℤ/2; ℤ/2) receives a map from Σ^{−1}H_{∗}(K(ℤ/2, 1); ℤ/2). Then ξ_{i} is the image of the generator of H_{2^{i}}(K(ℤ/2, 1); ℤ/2) ≅ ℤ/2. 1.5.3. The ξ_{i} are the indecomposable elements of A^{∗} (i.e., generate I(A^{∗})/I(A^{∗})^{2} where I(A^{∗}) ⊂ A^{∗} is the augmentation ideal ker(ϵ)), and are dual to the primitive elements Q_{i−1} in A (i.e., those x ∈ A such that ψ(x) = x ⊗ 1 + 1 ⊗ x). These Milnor primitives can be inductively defined by Q_{0} = Sq^{1} and Q_{n} = [Q_{n−1}, Sq^{2^{n}}] for n ≥ 1. 1.5.4. Directly dual to the A-module action λ on H^{∗}(X) is a homomorphism

λ^{∗}: H_{∗}(X) → H_{∗}(X) ⊗ A^{∗}

making H_{∗}(X) a (cocommutative coaugmented) right A^{∗} -comodule coalgebra. The two dualized viewpoints are related as follows. With I running through the admissible sequences the {Sq^{I}}_{I} form a basis for A. There is a dual basis {(Sq^{I})^{∗}}_{I} for A^{∗}, and the homology operations {Sq_{∗}^{I}}_{I} give a basis for A^{op} . Then ρ(x ⊗ Sq_{∗}^{I}) = Sq_{∗}^{I}(x) and

λ^{∗}(x) = ∑_{I} Sq_{∗}^{I}(x) ⊗ (Sq^{I})^{∗}

for x ∈ H_{∗}(X).

2. Infinite loop spaces

2.1. Iterated loop spaces. 2.1.1. Let n ≥ 1. A space X is an n-fold loop space if there is a space X_{n} and a homotopy equivalence X ≃ Ω^{n}X_{n}. We call X_{n} an n-fold delooping of X. Then for 0 ≤ k ≤ n there are spaces X_{k} = Ω^{n−k}X_{n}, with ΩX_{k+1} ≅ X_{k} for 0 ≤ k < n, and X ≃ X_{0} .

2.1.2. Any loop space is naturally an H -group, i.e., a group up to homotopy. If X ≃ ΩY then composition of loops defines a product

μ: X × X ≃ ΩY × ΩY ⟶ ΩY ≃ X

and the base point defines a unit map η: ∗ → X. These satisfy associativity and left and right unit axioms up to homotopy. Furthermore, reversing the orientation of a loop determines a homotopy inverse

ι: X ≃ ΩY ⟶ ΩY ≃ X

which acts as a left and right inverse, up to homotopy. Thus X represents a group valued functor Z ↦ [Z, X]. 2.1.3. A space X is an infinite loop space, or Ω^{∞} -space, if there is a space X_{n} for each n ≥ 0, with ΩX_{k+1} ≅ X_{k} for all k ≥ 0, and X ≃ X_{0} . The sequence of spaces (X_{n})_{n≥0} is called a spectrum, and X_{0} is its underlying space. Hence an infinite loop space is any space homotopy equivalent to the underlying space of a spectrum. See [May1]. 2.1.4. A spectrum E = (E_{n})_{n≥0} (with ΩE_{n+1} ≅ E_{n} for all n ≥ 0) represents a generalized cohomology theory Ẽ^{∗}, defined on CW-complexes X by

Ẽ^{n}(X) = [X, E_{n}]

(based homotopy classes) for all n ≥ 0. This is a contravariant homotopy functor with natural suspension isomorphisms Ẽ^{n}(X) ≅ Ẽ^{n+1}(ΣX) for all n and spaces X. Equivalently, Ẽ^{∗} satisfies excision, admits long exact sequences for pairs, etc. 2.1.5. Dually, E represents a generalized homology theory Ẽ_{∗}, defined on CWcomplexes by

Ẽ_{n}(X) = colim_{k} π_{n+k}(E_{k} ∧ X)

for all integers n. This is a covariant homotopy functor with natural suspension isomorphisms Ẽ_{n}(X) ≅ Ẽ_{n+1}(ΣX), as above. Again Ẽ_{∗} satisfies excision, admits long exact sequences for pairs, etc. The coefficient groups of the generalized homology theory are the groups Ẽ_{n} = Ẽ_{n}(S^{0}).

2.2. Examples of infinite loop spaces. 2.2.1. Infinite loop spaces are ubiquitous, i.e., constantly encountered. 2.2.2. For each Abelian group A and k ≥ 0, the Eilenberg–Mac Lane space K(A, k) is an infinite loop space. Its n-th delooping is a K(A, n + k)-space. The spectrum HA with n-th space K(A, n) has underlying space K(A, 0) ≃ A, and represents ordinary singular homology and cohomology with coefficients in A. 2.2.3. Let

Q(X) = colim_{n} Ω^{n}Σ^{n}X,

where Ω^{n}Σ^{n}X → Ω^{n+1}Σ^{n+1}X is obtained by applying Ω^{n} for Y = Σ^{n}X to the map Y → ΩΣY that takes a point y ∈ Y to the loop t ↦ y ∧ t in ΣY = Y ∧ S^{1} . Then Q(X) ≅ ΩQ(ΣX). Then Q(X) is the free infinite loop space on X, in the sense that the functor Q from spaces to infinite loop spaces is left adjoint to the forgetful functor mapping the other way. Equivalently, there is a natural bijective correspondence between infinite loop maps Q(X) → Y and space level maps X → Y , when Y is an infinite loop space. 2.2.4. One important example is QS^{0} = colim_{n} Ω^{n}S^{n}. It is the underlying space in the sphere spectrum S, with n-th space Q(S^{n}). It represents stable homotopy and cohomotopy as generalized homology and cohomology theories. Its coefficient groups π_{∗}QS^{0} are the stable homotopy groups of spheres. 2.2.5. By Bott periodicity, Ω^{2}(ℤ × BU) ≃ ℤ × BU, while Ω^{8}(ℤ × BO) ≃ ℤ × BO and Ω^{8}(ℤ × BSp) ≃ ℤ × BSp. Here BG denotes the classifying space of the group G, with U, O and Sp the infinite unitary, orthogonal and symplectic groups. Hence ℤ × BU is an infinite loop space, with n-th delooping homotopy equivalent to ℤ × BU for n ≥ 0 even, and Ω(ℤ × BU) ≃ U for n ≥ 1 odd. It is the underlying space of the complex K -theory spectrum KU, representing complex topological K -theory and K -homology as generalized cohomology and homology theories, respectively. Likewise ℤ×BO and ℤ×BSp are infinite loop spaces, underlying the spectra KO and KSp representing real, resp. quaternionic, topological K -theory. 2.2.6. The connected, 1-connected and 2-connected covers of ℤ × BO are BO, BSO and BSpin. The connected and 2-connected covers of ℤ × BU are BU and BSU. The connected cover of ℤ × BSp is BSp. Here SO and SU are the infinite special (determinant 1) orthogonal and unitary groups, and Spin is the infinite spin group, which is the universal (double) cover of SO.

2.3. The little cubes operad. 2.3.1. To discuss the geometry of n-fold loop spaces, or infinite loop spaces, we need to consider certain spaces of operations on such iterated loop spaces. These combine to form a structure called an operad, for which we omit the precise definition. See [May3]. These spaces were introduced by Boardman and Vogt. 2.3.2. Let I^{n} = I × ⋯ × I denote the unit n-cube. A little n-cube f in I^{n} is a linear embedding f: I^{n} → I^{n} with parallel axes, i.e., it is of the form f = f_{1} × ⋯ × f_{n} where each f_{s}: I → I is a linear function f_{s}(t) = (1 − t)x_{s} + ty_{s} with 0 ≤ x_{s} < y_{s} ≤ 1. Let C_{n}(j) be the set of those ordered j -tuples

c = ⟨c_{1}, …, c_{j}⟩

of little n-cubes whose images have pairwise disjoint interiors in I^{n}. Topologize C_{n}(j) as a subspace of the space of maps ∐^{j} I^{n} → I. This is the space of little n-cubes. Multiplication by the identity little 1-cube 1: I → I defines stabilization maps C_{n}(j) → C_{n+1}(j), replacing each little n-cube f by the little (n + 1)-cube f × 1. Let C_{∞}(j) = colim_{n} C_{n}(j) be the union of these spaces, i.e., the space of j little ∞-cubes. 2.3.3. As an example, consider f: I → I and g: I → I given by f(t) = t/2 and g(t) = (t + 1)/2. Then ⟨f, g⟩ is a point in C_{1}(2). We shall see in the next section that this point corresponds to the usual multiplication on loop spaces given by composition of loops. Similarly ⟨f × 1^{n−1}, g × 1^{n−1}⟩ is a little n-cube, representing the loop sum on n-fold loop spaces. 2.3.4. The symmetric group Σ_{j} acts on C_{n}(j) from the right, by permuting the ordering of the j little n-cubes. For example, the order two element T ∈ C_{2} ≅ Σ_{2} takes ⟨f, g⟩ to ⟨g, f⟩, which represents multiplication of loops in the opposite order. Then ⟨f, g⟩ and ⟨g, f⟩ are in different components of C_{1}(2), but their images in C_{n}(2) are in the same component for all n ≥ 2. This reflects the homotopy commutativity of loop sum on n-fold loop spaces for n ≥ 2, and that π_{n}(X) is Abelian for n ≥ 2. 2.3.5. Let F (ℝ^{n}; j) be the configuration space of j distinct points (x_{1}, …, x_{j}) in ℝ^{n}. Then Σ_{j} acts freely on F (ℝ^{n}; j) by permuting the points, and F (ℝ^{n}; j) is (n − 2)-connected. The inclusion ℝ^{n} → ℝ^{n+1} induces Σ_{j} -equivariant maps F (ℝ^{n}; j) → F (ℝ^{n+1}; j), with union

F (ℝ^{∞}; j) = colim_{n} F (ℝ^{n}; j).

Then F (ℝ^{∞}; j) is a free contractible Σ_{j} -space, hence has the Σ_{j} -equivariant homotopy type of EΣ_{j}. 2.3.6. There is a Σ_{j} -equivariant homotopy equivalence C_{n}(j) → F (ℝ^{n}; j), which takes a j -tuple ⟨c_{1}, …, c_{j}⟩ to the configuration (x_{1}, … x_{j}) of center points

x_{s} = c_{s}(1/2, …, 1/2)

of the little n-cubes. Hence C_{∞}(j) ≃ EΣ_{j} for all j ≥ 0. 2.3.7. When j = 2, C_{n}(2) is the space of pairs ⟨c_{1}, c_{2}⟩ of little cubes in I^{n} with disjoint interiors, and T ∈ C_{2} acts by transposing their numbering. By the center point map, this is equivariantly homotopy equivalent to the configuration space F (ℝ^{n}; 2) of pairs of disjoint points (x_{1}, x_{2}) in ℝ^{n}, and T again transposes their numbering. This is in turn equivariantly homotopy equivalent to S^{n−1} with the antipodal action, by the map taking (x_{1}, x_{2}) to the unit vector pointing from x_{1} to x_{2} . These identifications are all compatible with stabilization in n, and provides an equivariant homotopy equivalence from C_{∞}(2) to S^{∞}, which is our preferred model for EC_{2} .

2.4. The action on iterated loop spaces. 2.4.1. Let X be an n-fold loop space, with n-th delooping X_{n}. So X ≃ Ω^{n}X_{n}. Assume that in fact X is an n-fold loop space in the strict sense, i.e., that X = Ω^{n}X_{n}. We will show that each point in C_{n}(j) defines a map

X^{j} = X × ⋯ × X ⟶ X,

i.e., an operation on X with j inputs and 1 output. In the next section we will use this map to produce homology operations from H_{∗}(X) to itself. Thereafter we may replace X by any homotopy equivalent space, and hence obtain homology operations also for n-fold loop spaces in the weaker sense, with X ≃ Ω^{n}X_{n}. 2.4.2. For c = ⟨c_{1}, … c_{j}⟩ ∈ C_{n}(j) and X = Ω^{n}Y , define a map

θ_{n,j}(c): X^{j} ⟶ X

as follows. Take points x_{1}, …, x_{j} ∈ X = Ω^{n}Y , viewed as maps y_{s}: I^{n} → S^{n} → Y . (Here I^{n} → S^{n} collapses ∂I^{n} ⊂ I^{n} to the base point, and identifies I^{n}/∂I^{n} ≅ S^{n} by a standard homeomorphism. For example, fix a homeomorphism I^{1}/∂I^{1} ≅ S^{1} and smash with itself n times.) Define θ_{n,j}(c): I^{n} → Y to be y_{s} ∘ c_{s}^{−1} on the subspace c_{s}(I^{n}) ⊆ I^{n}, for all 1 ≤ s ≤ j, and map the remainder of I^{n} to the base point. The resulting map I^{n} → Y takes ∂I^{n} to the base point, hence descends to a map S^{n} → Y , or equivalently a point in X = Ω^{n}Y . We rephrase the definition. A j -tuple of little n-cubes ⟨c_{1}, …, c_{j}⟩ and a j -tuple of maps (I^{n}, ∂I^{n}) → (Y, ∗) are paired up to define a map from the union

⋃_{s=1}^{j} c_{s}(I^{n}) ⊆ I^{n}

to Y , which is extended by mapping the complement to the base point. This defines a map (I^{n}, ∂I^{n}) → (Y, ∗). Thinking of such maps as points in X = Ω^{n}Y defines θ_{n,j}(c)(x_{1}, …, x_{j}). 2.4.3. Letting c = ⟨c_{1}, …, c_{j}⟩ vary, we obtain a continuous action map

θ_{n,j}: C_{n}(j) × X^{j} ⟶ X,

which is natural in n-fold loop spaces X = Ω^{n}Y . Each point in C_{n}(j) thus parametrizes a map X^{j} → X. The permutation group Σ_{j} acts diagonally on the left hand side, by renumbering j -tuples of little n-cubes in C_{n}(j) and by permuting the factors in X^{j}. The map θ_{n,j} ignores this action, and hence descends to a continuous map

θ′_{n,j}: C_{n}(j) ×_{Σ_{j}} X^{j} ⟶ X.

2.4.4. Now suppose X is an infinite loop space in the strict sense, i.e., that there are spaces (X_{n})_{n≥0} with X = X_{0} and X_{n} ≅ ΩX_{n+1} for all n ≥ 0. Then the maps θ_{n,j} and θ_{n+1,j} are compatible with the stabilization C_{n}(j) → C_{n+1}(j), and so we obtain maps

θ_{∞,j}: C_{∞}(j) × X^{j} ⟶ X

and

θ′_{∞,j}: C_{∞}(j) ×_{Σ_{j}} X^{j} ⟶ X.

Combined with the Σ_{j} -equivariant homotopy equivalence EΣ_{j} ≃ C_{∞}(j), we get the structure map

θ_{j}: X_{hΣ_{j}}^{j} = EΣ_{j} ×_{Σ_{j}} X^{j} ⟶ X.

When j = 2 we may use S^{∞} with the antipodal action as our model for EΣ_{2} , and get a map

θ_{2} : S^{∞} ×_{C_{2}} (X × X) ⟶ X

where T ∈ C_{2} transposes the factors in X × X.

2.5. A monad. 2.5.1. Let X be any space. Define

C_{∞}X = ∐_{j≥0} C_{∞}(j) ×_{Σ_{j}} X^{j}/ ∼

where (d_{i}(c), x) ∼ (c, d_{i}(x)) for all c = ⟨c_{1}, …, c_{j}⟩, x = (x_{1}, …, x_{j−1}) and 0 ≤ i ≤ j. Here d_{i}(c) omits the i-th little cube c_{i} from c, while d_{i}(x) inserts the base point ∗ in the i-th position of x, shifting x_{i}, …, x_{j−1} one step to the right. Then C_{∞} is a monad, in the sense that there is a product μ: C_{∞}(C_{∞}X) → C_{∞}X induced by composition of little ∞-cubes, and a unit η: X → C_{∞}X mapping to the (j = 1)-summand. These satisfy associativity and unit axioms. 2.5.2. There is a natural map for any X

α_{∞}: C_{∞}X ⟶ Q(X) = colim_{n} Ω^{n}Σ^{n}X.

When X is an infinite loop space, the structure maps θ′_{∞,j} respect the identifications in C_{∞}X and combine to define a map

θ: C_{∞}X ⟶ X.

Then θ is the composite of α_{∞} with the counit map Q(X) → X obtained from the infinite loop space structure on X. The map θ satisfies transitivity and unit axioms with respect to the monad product μ. We say that X is a C_{∞} -algebra.

2.5.3. Approximation theorem (May). α_{∞}: C_{∞}X → Q(X) is a group completion for every space X. In particular, for X path connected it is a weak homotopy equivalence.

2.5.4. Recognition principle (May). Every infinite loop space is a C_{∞} -algebra, and every connected C_{∞} -algebra has the weak homotopy type of an infinite loop space.

See [May3] and [May4].

3. Homology of infinite loop spaces

3.1. The Pontryagin product. 3.1.1. An H -space X is a space with a homotopy associative and unital map μ: X × X → X. Then μ induces a product

ϕ: H_{∗}(X) ⊗ H_{∗}(X) ≅ H_{∗}(X × X) ⟶^{μ_{∗}} H_{∗}(X)

which, together with the unit

η: ℤ/2 = H_{∗}(∗) ⟶ H_{∗}(X)

induced by the base point inclusion i: ∗ → X, makes H_{∗}(X) an algebra. We call ϕ the Pontryagin product induced by the H -space structure map μ.

3.1.2. For an H -space X, the algebra and coalgebra structures on H_{∗}(X) induced by the Pontryagin product and the diagonal map are compatible, in the sense that the algebra product and unit are coalgebra homomorphisms, and dually the coalgebra coproduct and counit are algebra homomorphisms. This structure on H_{∗}(X) is called a Hopf algebra. Since the diagonal is cocommutative, H_{∗}(X) is always a cocommutative Hopf algebra. 3.1.3. When the H -space X admits a homotopy inverse ι: X → X, we call X an H -group. The inverse induces a conjugation

χ: H_{∗}(X) ⟶^{ι_{∗}} H_{∗}(X)

which satisfies ϕ(1 ⊗ χ)ψ = ηϵ. 3.1.4. Loop spaces X = ΩX_{1} provide examples of H -groups. The product μ: ΩX_{1} × ΩX_{1} → ΩX_{1} is given by loop sum, the unit η: ∗ → ΩX_{1} maps to the constant loop, and the inverse ι: ΩX_{1} → ΩX_{1} reverses the parametrization of a loop. Then H_{∗}(X) = H_{∗}(ΩX_{1}) is a cocommutative Hopf algebra with conjugation. We will write ab = ϕ(a ⊗ b) for the Pontryagin product on H_{∗}(X) when X is a loop space. 3.1.5. Second loop spaces X = Ω^{2}X_{2} provide examples of homotopy commutative H -groups. This follows since the space of 2 little n-cubes C_{n}(2) is connected for n ≥ 2, and so the multiplication maps μ and μT are homotopic. (Here T: X×X → X × X denotes the twist map.) Hence H_{∗}(X) is a commutative and cocommutative Hopf algebra with conjugation, when X is an n-fold loop space, with n ≥ 2. This applies in particular when X is an infinite loop space.

3.2. Structure theorems for Hopf algebras. 3.2.1. For a discussion of algebras, coalgebras, Hopf algebras and the associated module structures, see [MM]. That paper also proves the following structure theorems for Hopf algebras, due to Hopf, Leray and Borel. 3.2.2. Let A be a graded Hopf algebra over ℤ/2, with product ϕ, unit and coaugmentation η, coproduct ψ, and counit and augmentation ϵ. A Hopf algebra A is connected if the unit map η is an isomorphism in degree 0. It is of finite type if the underlying vector space in each degree is finite dimensional. The height of an element x ∈ A is the least integer q such that x^{q} = 0; or, if no such integer exists, the height of x is infinity.

3.2.3. Theorem (Borel). If A is a commutative connected Hopf algebra of finite type over ℤ/2, then there is an isomorphism of algebras

A ≅ ⨂_{i∈I} A_{i}

where each A_{i} is a Hopf algebra with a single algebra generator x_{i}.

3.2.4. Proposition. If A is a connected Hopf algebra over ℤ/2 which has one algebra generator x, then the height of x is either a power of two or infinity. Thus A ≅ ℤ/2[x]/(x^{2^{e}}) or A ≅ ℤ/2[x] as algebras.

3.2.5. Let I(A) = ker(ϵ) be the augmentation ideal, and dually let J(A) = cok(η). There are direct sum decompositions A ≅ I(A) ⊕ ℤ/2 and A ≅ ℤ/2 ⊕ J(A) since ϵη = 1 on ℤ/2. Let the vector space

P (A) = {x ∈ A | ψ(x) = x ⊗ 1 + 1 ⊗ x}

be the primitives in A, and let

Q(A) = I(A)/I(A) · I(A)

be the indecomposables of A. We think of P (A) as a subspace of A, and Q(A) as a quotient space of A. We say that A is primitively generated if it is generated by P (A) as an algebra.

3.2.6. Theorem. If A is a commutative, primitively generated, connected Hopf algebra of finite type over ℤ/2, then there is an isomorphism of Hopf algebras

A ≅ ⨂_{i∈I} A_{i}

where each A_{i} is a Hopf algebra with a single algebra generator x_{i}.

See [MM]. 3.2.7. When A has one algebra generator x as in Proposition 3.2.4, ψ(x) = x ⊗ 1 + 1 ⊗ x for degree reasons, and this determines the Hopf algebra structure on A in either of the cases A = ℤ[x]/(x^{2^{e}}) and A = ℤ/2[x].

3.3. The Araki–Kudo/Dyer–Lashof operations. 3.3.1. Araki and Kudo [AK] defined homology operations Q^{i} in the mod 2 homology of infinite loop spaces. Their work was extended by Browder in [Br]. Dyer and Lashof [DL] introduced the corresponding operations in mod p homology, for p an odd prime. The algebra of operations is known as the Dyer–Lashof algebra, and we will refer to the Q^{i} as Dyer–Lashof operations, in spite of the fact that we restrict to the mod 2 case, where the operations were first defined by Araki and Kudo. The mod 2 Steenrod operations are also known as reduced squares, as they take values in degrees below the degree of the cup product square. Dually the mod 2 Dyer–Lashof operations are increased squares, as they take values in degrees above the degree of the Pontryagin product square. 3.3.2. We extend our notations from section 1.2. Let C_{2} = {1, T } be the group of order 2. The space of 2 little ∞-cubes C_{∞}(2) is a free contractible C_{2} -space, and serves as a model for EC_{2} . The augmented singular complex ϵ: S_{∗}(C_{∞}(2)) → ℤ/2 is a C_{2} -free resolution of ℤ/2.

Also let ϵ: W_{∗} → ℤ/2 be the periodic C_{2} -free resolution of ℤ/2 with W_{i} = ℤ/2[C_{2}]{e_{i}} and ∂(e_{i+1}) = 1 · e_{i} + T · e_{i} for all i ≥ 0. There is a chain homotopy equivalence W_{∗} → S_{∗}(C_{∞}(2)) over ℤ/2. Finally there is a chain homotopy equivalence H_{∗}(X) → S_{∗}(X), where H_{∗}(X) is viewed as a chain complex with zero boundary maps, given by choosing a basis for H_{∗}(X) and choosing representing cycles in Z_{∗}(X) ⊂ S_{∗}(X) for these homology classes. 3.3.3. Now suppose X is a strict infinite loop space. Thus there are spaces X_{n} and homeomorphisms X ≅ Ω^{n}X_{n} for all n ≥ 0. We obtained operad actions

θ′_{∞,2} : C_{∞}(2) ×_{C_{2}} X^{2} → X

in 2.4.4. There are isomorphisms

H_{∗}(W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2}) ≅ H_{∗}(S_{∗}(C_{∞}(2)) ⊗_{C_{2}} S_{∗}(X)^{⊗2}) ≅ H_{∗}(C_{∞}(2) ×_{C_{2}} X^{2})

induced by the chain equivalences W_{∗} ≃ S_{∗}(C_{∞}(2)) and H_{∗}(X) ≃ S_{∗}(X), and the shuffle homomorphism. Hence we get a homomorphism

θ_{∗}: H_{∗}(W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2}) ⟶ H_{∗}(X)

by composing the isomorphisms above with (θ′_{∞,2})_{∗}. 3.3.4. Now suppose given a class x ∈ H_{n}(X). When i ≥ n we form

Q^{i}(x) = θ_{∗}(e_{i−n} ⊗ x ⊗ x)

in H^{n+i}(X). This makes sense, because e_{i−n} ⊗ x ⊗ x defines a cycle in W_{∗} ⊗_{C_{2}} H_{∗}(X)^{⊗2} . Therefore, this formula defines the Araki–Kudo/Dyer–Lashof squaring operation

Q^{i}: H_{n}(X) → H_{n+i}(X)

for i ≥ n = deg(x). When i < n we let Q^{i}(x) = 0. 3.3.5. For example, Q^{n}(x) = x^{2} when x = deg(n) is the Pontryagin square, since θ_{∗}(e_{0} ⊗ x ⊗ x) = ϕ(x ⊗ x). Next, Q^{n+1}(x) = θ_{∗}(e_{1} ⊗ x ⊗ x) with n = deg(x) is represented by the closed path from x^{2} = ϕ_{∗}(x ⊗ x) to itself, parametrized by a path in C_{∞}(2) from the point μ representing loop sum to the (antipodal) point μT representing loop sum in the opposite order.

3.3.6. Theorem. Let X be an infinite loop space. There are homomorphisms Q^{i}: H_{∗}(X) → H_{∗}(X) for all n, i ≥ 0, which satisfy:

(1) The Q^{i} are natural with respect to maps f: X → Y of infinite loop spaces.

(2) Q^{i} raises degrees by i.

(3) Q^{i}(x) = 0 if i < deg(x).

(4) Q^{i}(x) = x^{2} if i = deg(x), where x^{2} = ϕ(x ⊗ x) is the Pontryagin product square.

(5) Q^{i}(1) = 0 for all i > 0, where 1 = [∗] ∈ H_{0}(X) is the algebra unit element.

(6) The external, internal and diagonal Cartan formulas hold:

Q^{k}(x × y) = ∑_{i+j=k} Q^{i}(x) × Q^{j}(y)

in H_{∗}(X × Y) for x ∈ H_{∗}(X), y ∈ H_{∗}(Y),

Q^{k}(xy) = ∑_{i+j=k} Q^{i}(x)Q^{j}(y)

in H_{∗}(X) for x, y ∈ H_{∗}(X), and

ψ(Q^{k}(x)) = ∑_{i+j=k} ∑ Q^{i}(x′) ⊗ Q^{j}(x′′)

if ψ(x) = ∑ x′ ⊗ x′′, x ∈ H_{∗}(X).

(7) The Q^{i} are stable and the Kudo transgression theorem holds: Q^{i}σ_{∗}(x) = σ_{∗}Q^{i}(x) where σ_{∗}: H̃_{n−1}(ΩX) → H̃_{n}(X) is the homology suspension. If X is simply-connected and if x ∈ H_{n}(X) transgresses to y ∈ H_{n−1}(ΩX) in the Serre spectral sequence of the path space fibration ΩX → P X → X, then Q^{i}(x) ∈ H_{n+i}(X) transgresses to Q^{i}(y) ∈ H_{n+i−1}(ΩX).

(8) The Adem relations hold: If a > 2b then

Q^{a}Q^{b} = ∑_{i}(2i − a, a − b − i − 1)Q^{a+b−i}Q^{i}.

The binomial coefficient can also be written \binom{i−b−1}{2i−a}. The sum runs over i with a/2 ≤ i ≤ a − b − 1.

(9) The Nishida relations hold: Recall that Sq_{∗}^{a} is the homology operation dual to Sq^{a}. Then

Sq_{∗}^{a}Q^{b} = ∑_{i}(a − 2i, b − 2a + 2i)Q^{b−a+i}Sq_{∗}^{i}.

The binomial coefficient can also be written \binom{b−a}{a−2i}. The sum runs over i with a − b/2 ≤ i ≤ a/2.

For a proof, see [May5, §1]. Sq_{∗}^{1}Q^{b} = (b − 1)Q^{b−1} is a useful special case of the Nishida relations. 3.3.7. The composite Dyer–Lashof operation Q^{a}Q^{b}(x) is defined by the upper composite in the following diagram, where we briefly write θ_{j} for θ′_{∞,j}:

\xymatrix{C_{∞}(2) ×_{C_{2}} (C_{∞}(2) ×_{C_{2}} X^{2})^{2} \ar[rr]^{1×_{C_{2}} (θ_{2})^{2}} \ar[d]^{≃}_{γ} &  & C_{∞}(2) ×_{C_{2}} X^{2} \ar[d]^{θ_{2}} \\ C_{∞}(4) ×_{C_{2}⋉C_{2}^{2}} X^{4} \ar[r]^{π} & C_{∞}(4) ×_{Σ_{4}} X^{4} \ar[r]^{θ_{4}} & X}

The diagram commutes. Note the covering map π, which is induced by the inclusion C_{2} ⋉ C_{2}^{2} ⊂ Σ_{4} of a Sylow 2-subgroup in Σ_{4} . The induced homomorphism H_{∗}(B(C_{2} ⋉ C_{2}^{2})) → H_{∗}(BΣ_{4}) is a surjection, split by a transfer map. Hence the elements in the kernel of this map will correspond to relations among the Q^{a}Q^{b}. These are the Adem relations. 3.3.8. The Dyer–Lashof operations Q^{i} commute with the conjugation χ in the homology of an infinite loop space. This follows by stability 3.3.6(7) and naturality with respect to the inverse map ι on X ≅ ΩX_{1} .

3.4. The Dyer–Lashof algebra R. 3.4.1. Consider sequences I = (i_{1}, …, i_{k}) of integers with i_{s} ≥ 0. Define the degree, length and excess of I as before by

d(I) = ∑_{s=1}^{k} i_{s},  ℓ(I) = k  and

e(I) = i_{k} − ∑_{s=2}^{k}(2i_{s} − i_{s−1}) = i_{1} − ∑_{s=2}^{k} i_{s}.

The sequence I determines the homology operation

Q^{I} = Q^{i_{1}} ∘ ⋯ ∘ Q^{i_{k}}: H_{∗}(X) → H_{∗}(X)

which increases degrees by d(I). I is said to be admissible if i_{s} ≤ 2i_{s+1} for 1 ≤ s < k. The empty sequence I = () is admissible and satisfies d(I) = 0, ℓ(I) = 0 and e(I) = ∞; it determines the identity homology operation Q^{()} = 1. 3.4.2. Let F denote the free (associative, unital) algebra generated by {Q^{i}}_{i≥0} . For q ≥ 0 define J(q) to be the two-sided ideal of F generated by the Adem relations 3.3.6(8), and by the relations Q^{I} = 0 if e(I) < q. Define R(q) to be the quotient algebra F/J(q), and observe that there are successive quotient maps R(q) → R(q + 1). Here Q^{0} ≠ 1. Let R = R(0); R will be called the mod 2 Dyer–Lashof algebra. Then J(q) coincides with the two-sided ideal K(q) of elements in F that annihilate every homology class of degree ≥ q of every infinite loop space. Hence R(q) equals the algebra generated by the Dyer–Lashof operations of homology operations in degrees ≥ q on infinite loop spaces. 3.4.3. The admissible monomials Q^{I} of excess e(I) ≥ q form a ℤ/2-module basis for R(q). For R = R(0) these begin:

1, Q^{0}, Q^{0}Q^{0}, …; Q^{1}; Q^{2}, Q^{1}Q^{1}; Q^{3}, Q^{2}Q^{1}; Q^{4}, Q^{2}Q^{2}, Q^{2}Q^{1}Q^{1};

Q^{5}, Q^{3}Q^{2}; Q^{6}, Q^{4}Q^{2}, Q^{3}Q^{3}, Q^{3}Q^{2}Q^{1}; Q^{7}, Q^{4}Q^{3}, Q^{4}Q^{2}Q^{1}; …

3.4.4. The homology of an infinite loop space X is a left module over R: The pairing

R ⊗ H_{∗}(X) → H_{∗}(X)

takes Q^{i} ⊗ x to Q^{i}(x), and more generally Q^{I} ⊗ x to Q^{I}(x). This left R-module is allowable in the sense that J(q)H_{q}(X) = 0 for all q ≥ 0. 3.4.5. The Dyer–Lashof algebra R admits a unique right A-module structure such that the Nishida relations 3.3.6(9) are satisfied. The coproduct ψ defined on generators by

ψ(Q^{k}) = ∑_{i+j=k} Q^{i} ⊗ Q^{j}

and the counit ϵ with ϵ(Q^{0}) = 1 make R into a Hopf algebra, as well as a right A-module coalgebra.

3.4.6. A component coalgebra C is a coaugmented coalgebra which is a direct sum of connected coalgebras. Let

πC = {g ∈ C | ψ(g) = g ⊗ g, g ≠ 0}.

Then πC is a basis for C_{0} . A component Hopf algebra B is monomial if πB is a monoid under the product of B. If X is a based space, H_{∗}(X) = ⨁_{g∈π_{0}(X)} H_{∗}(X_{g}) is a component coalgebra, where X_{g} ⊆ X is the path component g ∈ π_{0}(X). The base point determines the coaugmentation η: ℤ/2 → H_{∗}(X). In this case, πH_{∗}(X) ≅ π_{0}(X). The homology of an H -space is a monomial component Hopf algebra. 3.4.7. An allowable R-algebra is a commutative allowable left R-module algebra such that Q^{i}(x) = x^{2} if i = deg(x). An allowable R-coalgebra is a cocommutative component allowable left R-module coalgebra. An allowable R-Hopf algebra (with conjugation) is a monoidal left R-module Hopf algebra (with conjugation) which is allowable both as an R-algebra and as an R-coalgebra. For any of these structures, an allowable AR-structure is an allowable left Rstructure and an unstable right A-structure of the same type, such that the Aand R-operations satisfy the Nishida relations. Here an unstable right A-structure in homology is the dual of an unstable left A-structure in cohomology. See [May5, §2].

3.5. Some free functors. 3.5.1. Consider the following categories:

\xymatrix{Allowable AR-Hopf algebras with conjugation \ar[d]^{forget} \\ Allowable AR-Hopf algebras \ar[d]^{forget} \ar[u]^{G} \\ Allowable AR-coalgebras \ar[d]^{forget} \ar[u]^{W} \\ Cocommutative component unstable right A-module coalgebras \ar[u]^{E}}

The homology of an infinite loop space lives in the uppermost category. There are forgetful functors going down the list. These admit left adjoints G, W and E, i.e., functors going up the list, yielding free objects in the respective categories. 3.5.2. We now describe the effect of the free functor E on the prime example of an object in the lowermost category, namely H_{∗}(X) for a based space X. Let JH_{∗}(X) = cok(η) be the quotient of H_{∗}(X) by ℤ/2 = H_{∗}(∗), which is isomorphic to H̃_{∗}(X). Then

EH_{∗}(X) = ℤ/2 ⊕ ⨁_{q≥0} R(q) ⊗ JH_{q}(X).

Here EH_{∗}(X) is a free allowable AR-coalgebra, with left R-action induced by the surjections ϵ: R → ℤ/2 and R → R(q). The coproduct is defined using the diagonal Cartan formula 3.3.6(6). As an example, EH_{∗}(S^{0}) = ℤ/2 ⊕ R. 3.5.3. Next we describe the free functor W on the object EH_{∗}(X). Note that JEH_{∗}(X) = cok(η) is the sum ⨁_{q≥0} R(q) ⊗ JH_{q}(X). Then

W EH_{∗}(X) = P (⨁_{q≥0} R(q) ⊗ JH_{q}(X))/K

where P (−) denotes the polynomial algebra generated by its argument, and K is the two-sided ideal generated by the relations Q^{i}(x) = x^{2} for i = deg(x). Then W EH_{∗}(X) is a free allowable AR-Hopf algebra. The product is that obtained from the polynomial algebra, the coproduct is determined by that of EH_{∗}(X). The A- and R-module actions are determined by the internal Cartan formulas 1.2.6(7) and 3.3.6(7). For example, W EH_{∗}(S^{0}) = P (R)/K is the polynomial algebra on the admissible Dyer–Lashof operations Q^{0} and Q^{I} of excess e(I) > 0, as an algebra. Its components are ℕ_{0} = {0, 1, 2, … }, with (Q^{0})^{k} in the k th component. 3.5.4. A connected allowable AR-Hopf algebra admits a unique conjugation. In general the free functor G to allowable AR-Hopf algebras with conjugation amounts to a localization, group completing the monoid πC of components. For example, when X is connected GW EH_{∗}(X) = W EH_{∗}(X). When X = S^{0} , GW EH_{∗}(S^{0}) is a sum indexed of integers k ∈ ℤ of copies of the polynomial algebra P (R(1)) on the admissible Q^{I} with e(I) > 0.

3.6 The homology of Q(X). 3.6.1. The homology of the space C_{∞}X from 2.5.1 also admits Dyer–Lashof operations, and is an allowable AR-Hopf algebra. The unit inclusion η: X → C_{∞}X induces a homomorphism η_{∗}: H_{∗}(X) → H_{∗}(C_{∞}X), whose left adjoint is a natural transformation

η̄_{∗}: W EH_{∗}(X) → H_{∗}(C_{∞}X)

extending η_{∗} over the inclusion H_{∗}(X) → W EH_{∗}(X).

3.6.2 Theorem (May). For every space X, η̄_{∗}: W EH_{∗}(X) → H_{∗}(C_{∞}X) is a natural isomorphism of allowable AR-Hopf algebras.

3.6.3. The homology of the free infinite loop space Q(X) from 2.2.3 certainly admits Dyer–Lashof operations, and is an allowable AR-Hopf algebra with conjugation. The unit inclusion η: X → Q(X) now induces a homomorphism η_{∗}: H_{∗}(X) → H_{∗}(Q(X)), whose left adjoint is a natural transformation

η̃_{∗}: GW EH_{∗}(X) → H_{∗}(Q(X))

extending η_{∗} over the inclusion H_{∗}(X) → GW EH_{∗}(X).

3.6.4 Theorem (Dyer–Lashof, May). For every space X, η̃_{∗}: GW EH_{∗}(X) → H_{∗}(Q(X)) is a natural isomorphism of allowable AR-Hopf algebras with conjugation.

3.6.5 Corollary (Nakaoka).

⨁_{n≥0} H_{∗}(BΣ_{n}) ≅ H_{∗}(C_{∞}S^{0}) ≅ W EH_{∗}(S^{0}) ≅ P (R)/K.

Here a product Q^{I_{1}} ·. . .·Q^{I_{k}} of admissible monomials in R contributes to the summand H_{∗}(BΣ_{n}) where n = ∑_{s=1}^{k} 2^{ℓ(I_{s})} . So Q^{i} generates H_{i}(BΣ_{2}). Multiplication by Q^{()} induces the injections H_{∗}(BΣ_{n}) → H_{∗}(BΣ_{n+1}) for all n.

3.7 Proof of Theorem 3.6.4. 3.7.1. We consider the proof of 3.6.4, concerning H_{∗}(Q(X)). For simplicity, let us assume X is path connected. The general case involves additional bookkeeping to account for the various components. Let tX be a basis for JH_{∗}(X) ≅ H̃_{∗}(X), and let P {T X} by the polynomial algebra generated by the set

T X = {Q^{I}(x) | x ∈ tX, I admissible, e(I) > deg(x)}.

Then as algebras

GW EH_{∗}(X) = W EH_{∗}(X) ≅ P {T X}.

3.7.2. If X is (q − 1)-connected, q > 1, then η_{∗}: H_{∗}(X) → H_{∗}(Q(X)) is an isomorphism for ∗ < 2q, by Freudenthal’s stability theorem. Likewise the inclusion H_{∗}(X) → GW EH_{∗}(X) is an isomorphism for ∗ < 2q. Thus the theorem holds in degrees < 2q is X is (q − 1)-connected. We claim that if the theorem holds for the suspension ΣX in degrees < n, then it holds for X in degrees < n − 1. This will finish the proof, since for any q > 1 then the theorem for Σ^{q}X in degrees < 2q implies the theorem for X in degrees < q. Letting q grow to infinity, the theorem follows. 3.7.3. Let {E^{r}} = {E_{∗∗}^{r}}_{r} be the Serre spectral sequence in homology for the path space fibration

Q(X) ≅ ΩQ(ΣX) → P Q(ΣX) → Q(ΣX).

Then

E_{∗∗}^{2} = H_{∗}(Q(ΣX)) ⊗ H_{∗}(Q(X))

since the base Q(ΣX) is simply-connected. Clearly E_{∗∗}^{∞} = ℤ/2 concentrated in bidegree (0, 0). 3.7.4. Take tΣX = {Σ_{∗}x | x ∈ tX} as the basis for JH_{∗}(ΣX). Here Σ_{∗}: H̃_{∗}(X) ≅ H̃_{∗}(ΣX). Construct a model spectral sequence {′E^{r}} = {′E_{∗∗}^{r}}_{r} with

′E_{∗∗}^{r} = W EH_{∗}(ΣX) ⊗ W EH_{∗}(X).

The differentials of {′E^{r}} are specified by requiring {′E^{r}} to be a spectral sequence of differential bigraded algebras, such that if Q^{I}(x) ∈ T X then there is a transgressive differential

τ(Q^{I}(Σ_{∗}x)) = Q^{I}(x).

Then {′E^{r}} is isomorphic to a tensor product of spectral sequences of the form

E{y} ⊗ P {τ(y)}.

Here y runs through

{Q^{I}(Σ_{∗}x) | x ∈ tX, I admissible, e(I) > deg(x)}.

This is because to the eyes of {′E^{r}}, the base W EH_{∗}(ΣX) behaves like a tensor product of exterior algebras on these y. Clearly ′E_{∗∗}^{∞} = ℤ/2 concentrated in bidegree (0, 0). 3.7.5. By construction, there is a unique morphism of algebras f: ′E_{∗∗}^{2} → E_{∗∗}^{2} such that

f = η̃_{∗} ⊗ η̃_{∗}: W EH_{∗}(ΣX) ⊗ W EH_{∗}(X) → H_{∗}(Q(ΣX)) ⊗ H_{∗}(Q(X)).

By 3.3.6(7) and the definition of {′E_{∗∗}^{r}}_{r}, f induces a morphism of spectral sequences. Hence f is a morphism of first quadrant algebra spectral sequences with E^{2} -terms of the form (base) ⊗ (fiber), both converging to ℤ/2 in bidegree (0, 0). By the Moore–Zeeman comparison theorem, then, the hypothesis that f induces an isomorphism η̃_{∗}: W EH_{∗}(ΣX) → H_{∗}(Q(ΣX)) in degrees < n on the base implies that f induces an isomorphism η̃_{∗}: W EH_{∗}(X) → H_{∗}(Q(X)) in degrees < n − 1 on the fiber. This completes the proof. 3.7.6. In effect, the proof shows that for y ∈ H_{∗}(ΣX), the classes Q^{I}(y) in H_{∗}(Q(ΣX)) for e(I) ≥ deg(y) transgress to classes Q^{I}(x) in H_{∗}(Q(X)), for e(I) > deg(x), with x = τ(y). In addition, the differential algebra structure generates powers (Q^{I}(x))^{k} of these classes in H_{∗}(Q(X)), but these are now products of terms Q^{J}(x) with excess one less: e(J) ≥ deg(x).

4. Calculations

4.1 E_{∞} ring spaces. Let X be an infinite loop space. The little cubes action maps

θ_{∞,j}: C_{∞}(j) × X^{j} → j

satisfy certain compatibility conditions, such that they form a structure called an E_{∞} -action on X. We say that X is an E_{∞} -space, or a homotopy-everything space. In particular, an E_{∞} -space is a homotopy commutative H -space. For any chosen point of C_{∞}(2) determines a multiplication μ: X×X → X, which is unital, commutative and associative up to homotopy by the contractibility of each C_{∞}(j), j ≤ 3. However, the presence of the higher θ_{∞,j} and contractibility of C_{∞}(j) for higher j ensures that an E_{∞} -structure is a more restrictive notion than just a homotopy commutative H -space. All ‘higher coherence homotopies’ are also available. The structure maps above allow us to define the Pontryagin product and Dyer– Lashof operations on H_{∗}(X) when X is any E_{∞} -space. For example,

C_{∞}S^{0} = ∐_{j≥0} C_{∞}(j)/Σ_{j} ≃ ∐_{j≥0} BΣ_{j}

is such an E_{∞} -space. Its homology is

⨁_{j≥0} H_{∗}(BΣ_{j}) ≅ H_{∗}(C_{∞}S^{0}) ≅ W EH_{∗}(S^{0}) ≅ P (R)/K

where P (R) is the polynomial algebra on the Dyer–Lashof algebra R, and K is the two-sided ideal of relations generated by Q^{i}(x) = x^{2} for i = deg(x). The isomorphism takes Q^{I} ∈ R to the class Q^{I}([1]), where [1] ∈ H_{0}(BΣ_{1}) is the nonzero class. This example shows that E_{∞} -spaces need not be H -groups, i.e., they need not be grouplike H -spaces. However, any grouplike E_{∞} -space is an infinite loop space. E_{∞} -spaces are the homotopical generalization of Abelian monoids, and grouplike E_{∞} -spaces (=infinite loop spaces) are the homotopical generalization of Abelian groups. E_{∞} -spaces arise, for example, as the classifying space BC of a permutative category C, or more generally of a symmetric monoidal category. Grouplike E_{∞} -spaces arise as the group completions of such classifying spaces, which equals the K -theory space K(C) = ΓBC of the category. The homotopy groups of this infinite loop space are the K -groups of the category. We also encounter the homotopical generalization of semi-rings (no additive inverse assumed) and rings. The former arise as the classifying spaces of bi-permutative categories, or more generally of symmetric bi-monoidal categories. As an example, consider the category O of finite dimensional real inner product spaces. It admits two symmetric monoidal pairings: direct sum ⊕ and tensor product ⊗ of vector spaces. Its classifying space BO ≃ ∐_{n≥0} BO(n) admits two induced H -space structures: one induced by direct sum BO(n) × BO(m) → BO(n + m), and one by tensor product BO(n) × BO(m) → BO(nm). As a different example, points of QS^{0} are represented by maps f: S^{n} → S^{n} and g: S^{m} → S^{m} for n, m large. The loop sum using a suspension coordinate in the source defines one (additive) pairing on QS^{0} , but the smash product of such maps, f ∧ g: S^{n+m} → S^{n+m}, defines another (multiplicative) pairing. Such spaces, then, have two E_{∞} -structures, and thus two associated H -space structures. We think of one of these as additive and the other as multiplicative, and assume they are related by suitable distributivity conditions. This structure is known as an E_{∞} -ring space. The homology H_{∗}(X) of an E_{∞} -ring space is thus a Hopf algebra with Pontryagin product ∗ and Dyer–Lashof operations Q^{i} derived from the additive E_{∞} -structure, but also a Hopf algebra with Pontryagin product # and Dyer–Lashof operations Q̃^{i} derived from the multiplicative E_{∞} -structure. Note that we denote the multiplicative Dyer–Lashof operations with a tilde. The common coproduct and right A-module structure combines with these two related products to yield an (allowable) AR-Hopf bialgebra. There are formulas in [May7, §§1–3] expressing distributivity of # over the additive ∗ and Q^{i}, the mixed Cartan formula for Q̃^{i}(x ∗ y), and the mixed Adem relations for Q̃^{a}(Q^{b}(x)). In principle, they inductively reduce the determination of the multiplicative Pontryagin product # and Dyer–Lashof operations Q̃^{i} in an E_{∞} ring space X to their evaluation on additive R-algebra generators for H_{∗}(X).

4.2. The homology of SO, BO etc.. Calculations begin with O(1) = C_{2} = Σ_{2} . Its classifying space BO(1) ≃ ℝP^{∞} has H^{∗}(BO(1)) = P {x} with deg(x) = 1, and dually H_{∗}(BO(1)) = ℤ/2{e_{i} | i ≥ 1}, with e_{i} dual to x^{i}. The diagonal inclusion O(1)^{n} → O(n) induces maps BO(1)^{n} → BO(n) of classifying spaces. The induced map on cohomology

H^{∗}(BO(n)) → H^{∗}(BO(1))^{⊗n} ≅ P {x_{1}, … x_{n}}

is an injection, with image the symmetric polynomials in the x_{1}, …, x_{n}. Hence

H^{∗}(BO(n)) ≅ P {w_{1}, …, w_{n}}

with w_{i} the ith Stiefel–Whitney class, mapping to the ith elementary symmetric polynomial σ_{i}(x_{1}, …, x_{n}) by the injection above. Letting n → ∞, we have

H^{∗}(BO) ≅ P {w_{i} | i ≥ 1}.

The coproduct tied to Whitney sum is ψ(w_{k}) = ∑_{i+j=k} w_{i} ⊗ w_{j}. Dually, the map on homology

H_{∗}(BO(1))^{⊗n} → H_{∗}(BO(n))

is surjective, identifying H_{∗}(BO(n)) with the coinvariants of the n-fold tensor power of ℤ/2{e_{i}}_{i} under the permutation action. Thus

H_{∗}(BO(n)) ≅ ⨁_{d≤n} P {e_{1}, …, e_{n}}_{d}

where P {e_{1}, …, e_{n}}_{d} denotes the degree d part of the polynomial algebra. Letting n → ∞, we have

H_{∗}(BO) = P {e_{i} | i ≥ 1}

where each e_{i} is the image of e_{i} ∈ H_{1}(BO(1)) under the inclusion BO(1) → BO. Let Q_{0} = Sq^{1} and Q_{n} = [Q_{n−1}, Sq^{2^{n}}] be the Milnor primitives in the Steenrod algebra A. Recall that H^{∗}(K(ℤ/2, 2)) ≅ P {ι_{2}, Q_{n}(ι_{2}) | n ≥ 0} where ι_{2} is the fundamental class, and

Q_{n}(ι_{2}) = Sq^{2^{n}} ∘ ⋯ ∘ Sq^{1}(ι_{2}).

The Serre spectral sequences for the 1- and 2-connected coverings of BO give:

H^{∗}(BSO) = P {w_{i} | i ≥ 2}
H^{∗}(BSpin) = H^{∗}(BSO)/(w_{2}, Q_{n}(w_{2}) | n ≥ 0).

Dually

H_{∗}(BSO) = P {e_{2}, e_{3}, … }
H_{∗}(BSpin) = (?).

The Serre spectral sequences for the path space fibration for BSO and BSpin yields:

H^{∗}(SO) = P {σ^{∗}(w_{2i}) | i ≥ 1}
H^{∗}(Spin) = H^{∗}(SO)/(σ^{∗}(w_{2}))

where σ^{∗}(w_{2i}) in degree 2i − 1 transgresses to w_{2i}. Dually

H_{∗}(SO) = Γ{a′_{2i−1} | i ≥ 1}
H_{∗}(Spin) = Γ{a′_{2i−1} | i ≥ 2}

(divided power algebras) with a′_{2i−1} dual to σ^{∗}(w_{2i}). This gives a different basis for H_{∗}(SO), compared to

H_{∗}(SO) = E{a_{i} | i ≥ 1}

where a_{i} is the image of e_{i} under the reflection map BO(1) → SO taking a line L in ℝ^{∞} to the reflection in L followed by reflection in a fixed line.

4.3. The homology of SG, G/O and BSG. There is a unit map QS^{0} → BO × ℤ of E_{∞} ring spaces. One way to construct it is as an induced map of K -theory spaces: Let E be the symmetric bimonoidal category of finite sets, under disjoint union ∐ and Cartesian product ×. Likewise let O be the symmetric bimonoidal category of finite dimensional real inner product spaces, under direct sum ⊕ and tensor product ⊗. There is a symmetric bimonoidal functor E → O taking a finite set U to the real vector space V = ℝ{U} with U as an orthonormal basis. Then on classifying spaces we have a map

BE ≃ ∐_{n≥0} BΣ_{n} ⟶ BO ≃ ∐_{n≥0} BO(n)

of E_{∞} ring spaces. Group completing, we obtain the unit map

K(E) ≃ QS^{0} ⟶ K(O) ≃ BO × ℤ

by way of the Barratt–Priddy–Quillen equivalence on the left. On π_{0} , the unit map is an isomorphism ℤ ≅ ℤ. Let [i] ∈ H_{0}(QS^{0}) denote the homology class of a point in the degree i component Q_{i}(S^{0}) ⊂ QS^{0} . Restricting the unit map to the 0-components, we obtain an additive infinite loop map Q_{0}(S^{0}) → BO = BO_{⊕}. Restricting instead to the 1-components, we get a multiplicative infinite loop map SG = Q_{1}(S^{0}) → BO_{⊗}. (These connected E_{∞} -spaces are actually infinite loop spaces, as noted above.) Consider the infinite loop space fiber sequence

SO ⟶^{j} SG → G/O → BSO ⟶^{Bj} BSG.

Here j is the natural inclusion (the j -map), taking an isometry ℝ^{n} → ℝ^{n} in SO(n) to its induced map on 1-point compactifications S^{n} → S^{n} in SG. Then j is an infinite loop map from the additive infinite loop space structure on SO ≃ ΩBSO_{⊕} to the multiplicative infinite loop space structure on SG = Q_{1}(S^{0}). Next, Bj is the classifying map of j, and is G/O the homotopy fiber of Bj. The calculation of H_{∗}(QS^{0}) gives H_{∗}(Q_{0}(S^{0})) ≅ P (X_{0}) as an algebra under ∗, where

X_{0} = {Q^{I}([1]) ∗ [2 − 2^{ℓ(I)}] | I admissible, ℓ(I) ≥ 1 and e(I) > 0}.

We can define a shifted ∗-product \underline{∗} on H_{∗}(SG) by x\underline{∗}y = (x∗[−1])∗(y∗[−1])∗[1]. Then H_{∗}(SG) ≅ P (X_{1}) as an algebra under \underline{∗}, where

X_{1} = {Q^{I}([1]) ∗ [1 − 2^{ℓ(I)}] | I admissible, ℓ(I) ≥ 1 and e(I) > 0}.

However, we are interested in the multiplicative infinite loop space structure on SG, which is related to that on G/O, and therefore rather want to know the algebra structure of H_{∗}(SG) under #. For an admissible sequence I with d(I) > 0, define

x_{I} = Q^{I}([1]) ∗ [1 − 2^{ℓ(I)}] ∈ H_{∗}(SG).

In particular x_{i} = Q^{i}([1]) ∗ [−1] for i ≥ 1.

Theorem.

(1) H_{∗}(SO) ≅ E{a_{i} | i ≥ 1} as an algebra. Here a_{i} is the image of the generator in H_{i}(BO(1)) under the reflection map BO(1) → SO.

(2) H_{∗}(SG) ≅ E{x_{i} | i ≥ 1} ⊗ P (X) as an algebra under #, where

X = {x_{I} | ℓ(I) > 2 and e(I) > 0, or ℓ(I) = 2 and e(I) ≥ 0}.

Here X is the union of the x_{(i,i)} with i ≥ 1 corresponding to I = (i, i) of excess 0, and the x_{I} with ℓ(I) ≥ 2 and e(I) > 0.

(3) j_{∗}(a_{i}) = x_{i} for all i ≥ 1, so im(j_{∗}) = E{x_{i} | i ≥ 1} and

H_{∗}(G/O) ≅ H_{∗}(SG)//H_{∗}(SO) ≅ P (X).

(4) H_{∗}(BO) ≅ P {v_{i} | i ≥ 1}, where v_{i} is the image of the generator e_{i} in H_{i}(BO(1)) under the map BO(1) → BO.

(5) H_{∗}(BSO) ≅ P {v_{i} | i ≥ 2}, where the v_{i} ’s correspond under the map BSO → BO. Here v_{i+1} = σ_{∗}(a_{i}) where σ_{∗} is the homology suspension, so v_{i+1} transgresses to a_{i} ∈ H_{i}(SO) in the Serre spectral sequence for the path space fibration for BSO.

(6) H_{∗}(BG) ≅ H_{∗}(BO) ⊗ E{σ_{∗}(x_{(i,i)})} ⊗ P (BX) as a Hopf algebra. Here

BX = {σ_{∗}(x_{I}) | ℓ(I) > 2 and e(I) > 1, or ℓ(I) = 2 and e(I) ≥ 1}.

(7) H_{∗}(BSG) ≅ H_{∗}(BSO) ⊗ E{σ_{∗}(x_{(i,i)})} ⊗ P (BX) as a Hopf algebra, with BX as above.

See [May7, §5]. The various non-classical parts are due to Milgram, May and Madsen. Comparing the sets of polynomial generators X and X_{1} , note that passing from the product \underline{∗} to # we delete the elements x_{i} and adjoin their squares x_{(i,i)} under the \underline{∗} product. Thus the appearance of these generators in H_{∗}(SG) is forced by the relations x_{i}^{2} = 0, arising in H_{∗}(SG) since H_{∗}(SO) is an exterior algebra.

4.4. The R-algebra structure of H_{∗}(SG). In the previous section we described the multiplicative #-algebra structure on H_{∗}(SG) in terms of the generators x_{I}, but these were defined by means of the additive homology operations Q^{i}. We would rather have a description in terms of the multiplicative homology operations Q̃^{i}.

Theorem (Madsen). H_{∗}(SG) ≅ E{x_{i} | i ≥ 1} ⊗ P (X̃) as an algebra under #, where

X̃ = {Q̃^{J}(x_{K}) | ℓ(K) = 2 and x_{(J,K)} ∈ X }.

Both E{x_{i} | i ≥ 1} = H_{∗}(SO) and P (X̃) are sub AR-Hopf algebras of H_{∗}(SG). Thus H_{∗}(G/O) ≅ P (X̃).

For I = (J, K), ℓ(K) = 2, such that x_{I} ∈ X, write x̃_{I} = Q̃^{J}(x_{K}) for the corresponding element of X̃.

Corollary. As Hopf algebras

H_{∗}(BSG) ≅ H_{∗}(BSO) ⊗ E{σ_{∗}(x_{(i,i)}) | i ≥ 1} ⊗ P (BX̃)

and

H_{∗}(B(G/O)) = H_{∗}(BSG)//H_{∗}(BSO) ≅ E{σ_{∗}(x_{(i,i)}) | i ≥ 1} ⊗ P (BX̃)

where

BX̃ = {σ_{∗}(x̃_{I}) | ℓ(I) > 2 and e(I) > 1 or ℓ(I) = 2 and e(I) ≥ 1}.

Thus the admissible x_{i} and x_{K} with ℓ(K) = 2 generate H_{∗}(SG) as a (multiplicative) R-algebra. Furthermore, operations Q̃^{i}(x_{K}) with (i, K) inadmissible can decompose many of the x_{K} with ℓ(K) = 2:

Theorem (Madsen). The following set is a basis for the ℤ/2-module of Ralgebra indecomposable elements of H_{∗}(SG):

{x_{2^{k}} | k ≥ 0} ∪ {x_{(2^{k},2^{k})} | k ≥ 0} ∪ {x_{(2^{k}n+2^{k},2^{k}n)} | n ≥ 1 and k ≥ 0}

See [Mad] and [May7, §6, §13]. The set X̃ contains precisely one element from the set above in each degree ≥ 2.

4.5. Homology operations for orthogonal and unitary groups. The symmetric bi-monoidal category of real inner product spaces has classifying space BO ≃ ∐_{n≥0} BO(n), with group completion ΓBO ≃ BO × ℤ. We identify BO with its 0-component Γ_{0}BO ⊂ ΓBO. In H_{∗}(BO(1)) let {e_{i}} be the standard basis. Let

BO(1) ⟶^{η} ∐_{n≥0} BO(n) ⟶^{ι} BO × ℤ

be the standard inclusions, and let v_{i} = ι_{∗}η_{∗}(e_{i}) and v̄_{i} = v_{i} ∗ [−1] ∈ H_{i}(BO). Then H_{∗}(BO) ≅ P {v̄_{i} | i ≥ 1} as an algebra under ∗.

Theorem (Priddy).

Q^{a}(v_{b}) = ∑_{i}(a − b − 1, b − i)v_{i} ∗ v_{a+b−i}.

The binomial coefficient can also be written \binom{a−i−1}{b−i}.

This determines Q^{a}(v̄_{b}) via the Cartan formula

Q^{k}(v̄_{b}) = ∑_{i+j=k} Q^{i}(v_{b}) ∗ Q^{j}([−1]).

The Q^{j}([−1]) can be inductively determined by relating the Q^{i} to the conjugation χ. Modulo ∗ decomposable elements,

Q^{a}(v̄_{b}) ≡ (a − b − 1, b − i)v̄_{a+b},

but the precise decomposables appearing are not so easy to list. The symmetric bi-monoidal category of complex inner product spaces has classifying space BU ≃ ∐_{n≥0} BU(n), with group completion ΓBU ≃ BU ×ℤ. We identify BU with its 0-component Γ_{0}BU ⊂ ΓBU. Let f_{i} ∈ H_{i}(BU(1)) be the image of e_{i} under the complexification map BO(1) → BU(1). Then f_{2i−1} = 0 and {f_{2i} | i ≥ 0} is a basis for H_{∗}(BU(1)). Let η: BU(1) → ∐_{n≥0} BU(n) → BU ×ℤ be the standard inclusion, and let v_{i} = η_{∗}(f_{i}) and f̄_{i} = f_{i} ∗ [−1] ∈ H_{i}(BU). As before, v_{2i−1} = v̄_{2i−1} = 0. Then H_{∗}(BU) ≅ P {v̄_{2i} | i ≥ 1} as an algebra under ∗.

Theorem (Priddy).

Q^{2a}(v_{2b}) = ∑_{i}(a − b − 1, b − i)v_{2i} ∗ v_{2a+2b−2i}.

The binomial coefficient can also be written \binom{a−i−1}{b−i}.

Again this determines Q^{2a}(v̄_{2b}) via the Cartan formula. See [May7, §§7–8].

4.6. The complex image of J space, JU. The K -theory K(𝔽_{3}) of the category GL(𝔽_{3}) of finite dimensional 𝔽_{3} -vector spaces is the group completion of the classifying space BGL(𝔽_{3}) ≃ ∐_{n≥0} BGL_{n}(𝔽_{3} . It is homotopy equivalent to the homotopy fiber JU × ℤ of

ψ^{3} − 1: BU × ℤ → BU,

also known as the complex image of J space. We identify JU with the 0-component Γ_{0}BGL(𝔽_{3}). Consider the standard inclusions

BO(1) ⟶^{η} ∐_{n≥0} BGL_{n}(𝔽_{3}) ⟶^{ι} JU × ℤ.

The first map identifies O(1) and GL_{1}(𝔽_{3}). Let v_{i} = ι_{∗}η_{∗}(e_{i}) and v̄_{i} = v_{i} ∗ [−1] ∈ H_{i}(JU).

Theorem (Quillen).

H_{∗}(JU) ≅ P {v̄_{2i} | i ≥ 1} ⊗ E{v̄_{2i−1} | i ≥ 1}

as an algebra under ∗.

Again Priddy’s theorem above allows the determination of the operations Q^{i} on H_{∗}(JU), in much the same way as for H_{∗}(BO). The elements

p_{2i−1} = v̄_{2i−1} + ∑_{j=1}^{i−1} v̄_{2j} ∗ p_{2i−2j−1}

p_{4i} = iv̄_{2i} ∗ v̄_{2i} + ∑_{j=1}^{i−1} v̄_{2j} ∗ v̄_{2j} ∗ p_{4i−4j}

for i ≥ 1 form a basis for the primitive elements in H_{∗}(JU).

4.7. The orthogonal image of J -space, JO. The K -theory of the category O(𝔽_{3}) of finite dimensional inner product spaces over 𝔽_{3} might be called the orthogonal K -theory KO(𝔽_{3}) of the finite field. It is homotopy equivalent to the homotopy fiber JO × ℤ of

ψ^{3} − 1: BO × ℤ ⟶ BSO.

We identify JO ≅ Γ_{0}BO(𝔽_{3}). There is an involution Φ on the subcategory of even dimensional inner product spaces in O(𝔽_{3}), acting on 2 × 2-matrices by conjugation with γ = ( \begin{matrix}−1 & 1 \\ 1 & 1 \end{matrix} ). Thus Φ_{∗} is an involution on H_{∗}(JO).

Theorem.

H_{∗}(JO) ≅ P {v̄_{i} | i ≥ 1} ⊗ E{ū_{i} | i ≥ 1}

as an algebra under ∗, where (Φ − 1)_{∗}(v̄_{i}) = ū_{i}. The homology operations satisfy:

Q^{a}(v_{b}) = ∑_{i,j,k}(a − b − 1, b − j)v_{i} ∗ v_{k} ∗ ū_{j−i} ∗ ū_{a+b−j−k}
Q^{a}(ū_{b}) = ∑_{i,j}(a − b − i − 1, b − j)ū_{i} ∗ ū_{j} ∗ ū_{a+b−i−j}

4.8. The real image of J -space, J. The K -theory of the category N (𝔽_{3}) of finite dimensional inner product spaces over 𝔽_{3} with morphisms of spinor norm equal to their determinant is denoted KN(𝔽_{3}). It is homotopy equivalent to the homotopy fiber J × ℤ of

ψ^{3} − 1: BO × ℤ ⟶ BSpin.

We identify J ≅ Γ_{0}BN (𝔽_{3}). This is the real image of J space.

There is a unit map e: QS^{0} → J × ℤ of E_{∞} ring spaces, lifting the unit map QS^{0} → BO × ℤ. Write J_{⊗} for the 1-component of J × ℤ, then e: SG → J_{⊗} is an infinite loop map, related to the Adams e-invariant. Let C = hofib(e) be the homotopy fiber of e, called the cokernel of J space. Hence there is a fiber sequence of infinite loop spaces

C ⟶ SG ⟶^{e} J_{⊗}.

A solution to the Adams conjecture yields a map α: J → SG, such that the composite eα: J → J_{⊗} is a homotopy equivalence. Such a solution exists as a space level map, but not as an H -map. Hence as groups, π_{∗}(SG) ≅ π_{∗}(J)⊕π_{∗}(C), with the summand π_{∗}(J) closely related to the image of the j -homomorphisms j_{∗}: π_{∗}(SO) → π_{∗}(SG). This is why J is called the (real) image of J space, and C the cokernel of J space. Let ū_{0} = v̄_{0} = [0]. The two bases for H_{∗}(SO) in 4.2 are hiding behind a change of basis in the following theorem.

Theorem.

H_{∗}(J) ≅ P { ∑_{i+j=a} ū_{i} ∗ v̄_{j} | a ≥ 1} ⊗ E{ū′_{i} | i ≠ 2^{k}}

where ū′_{i} ∈ E{ū_{i}} and ū_{i} + ū′_{i} is decomposable under ∗.

Corollary. No H -map J → SG can induce a monomorphism on H_{2} .

Proof. H_{2}(J) contains (ū_{1} + v̄_{1})^{2} = v̄_{1}^{2} as an ∗-algebra, and H_{1}(SG) has basis {x_{1}} with x_{1}^{2} = 0 under the #-product, so v̄_{1}^{2} maps to 0 under any H -map. □

((More from [May7, §§12–13] ?))

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